Semi‐analytical solution of MHD flow through boundary integrals on the pipe wall

A mathematical model is given for the magnetohydrodynamic (MHD) pipe flow as an inner Dirichlet problem in a 2D circular cross section of the pipe, coupled with an outer Dirichlet or Neumann magnetic problem. Inner Dirichlet problem is given as the coupled convection‐diffusion equations for the velocity and the induced current of the fluid coupling also to the outer problem, which is defined with the Laplace equation for the induced magnetic field of the exterior region with either Dirichlet or Neumann boundary condition. Unique solution of inner Dirichlet problem is obtained theoretically reducing it into two boundary integral equations defined on the boundary by using the corresponding fundamental solutions. Exterior solution is also given theoretically on the pipe wall with Poisson integral, and it is unique with Dirichlet boundary condition but exists with an additive constant obtained through coupled boundary and solvability conditions in Neumann wall condition. The collocation method is used to discretize these boundary integrals on the pipe wall. Thus, the proposed procedure is an improved theoretical analysis for combining the solution methods for the interior and exterior regions, which are consolidated numerically showing the flow behavior. The solution is simulated for several values of problem parameters, and the well‐known MHD characteristics are observed inside the pipe for increasing values of Hartmann number maintaining the continuity of induced currents on the pipe wall.     

A numerical solution of the dirichlet problem on some special doubly connected regions

The aim of this paper is to give a convergence proof of a numerical method for the Dirichlet problem on doubly connected plane regions using the method of reflection across the exterior boundary curve (which is analytic) combined with integral equations extended over the interior boundary curve (which may be irregular with infinitely many angular points).     

Mixed exterior Laplace's problem

In [C. Amrouche, V. Girault, J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator. An approach in weighted Sobolev spaces, J. Math. Pures Appl. 76 (1997) 55-81], authors study Dirichlet and Neumann problems for the Laplace operator in exterior domains of Rn. This paper extends this study to the resolution of a mixed exterior Laplace's problem. Here, we give existence, unicity and regularity results in Lp's theory with 1<p<∞, in weighted Sobolev spaces.     

Exterior problems in the half-space for the Laplace operator in weighted Sobolev spaces

The purpose of this work is to solve exterior problems in the half-space for the Laplace operator. We give existence and unicity results in weighted Lp's theory with 1<p<∞. This paper extends the studies done in [C. Amrouche, V. Girault, J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator, an approach in weighted Sobolev spaces, J. Math. Pures Appl. 76 (1) (1997) 55-81] with Dirichlet and Neumann conditions.     

On the exterior two-dimensional Dirichlet problem for elliptic equations

A necessary and sufficient condition is given on the boundary datum in order to the Dirichlet problem for an elliptic equation in a two-dimensional exterior Lipschitz domain has a unique solution with a finite Dirichlet integral which converges uniformly at infinity to an assigned constant value.     

用单层位势求解尖角区域上的Dirichlet外问题

采用Kress变换以及处理第一类奇异核的积分方法,运用Nystrom方法利用单层位势求解尖角区域上的Dirichlet外问题.给出具体的算法和数值例子,通过数值例子可以看出用单层位势求解尖角区域上的Dirichlet外问题与用单双层结合求解所得的结果基本上一致,说明这种方法是有效的和可行的.     

Condensation of fermion pairs in a domain

We consider a gas of fermions at zero temperature and low density, interacting via a microscopic two-body potential which admits a bound state. The particles are confined to a domain with Dirichlet boundary conditions. Starting from the microscopic BCS theory, we derive an effective macroscopic Gross–Pitaevskii (GP) theory describing the condensate of fermion pairs. The GP theory also has Dirichlet boundary conditions. Along the way, we prove that the GP energy, defined with Dirichlet boundary conditions on a bounded Lipschitz domain, is continuous under interior and exterior approximations of that domain.     

Randwertaufgaben in der verallgemeinerten Potentialtheorie

The Neumann and Dirichlet boundary value problem of generalized potential theory is considered. Based on a compact imbedding result, existence and uniqueness theorems are obtained for Riemannian manifolds with compact boundary (‘interior and exterior domains’).     

On the low wave number asymtotics for the two-dimensional exterior dirichlet problem for the reduced wave equation

A new proof is given for the low wave number asymptotics for the solutions to the exterior Dirichlet problem for the reduced wave equation in two dimensions.     

On the Solution of the Exterior Dirichlet Problem for an Axisymmetric Potential

For an unbounded domain of the meridian plane with bounded smooth boundary that satisfies certain additional conditions, we develop a method for the reduction of the Dirichlet problem for an axisymmetric potential to Fredholm integral equations. In the case where the boundary of the domain is a unit circle, we obtain a solution of the exterior Dirichlet problem in explicit form.     

The Exterior Dirichlet Problem for the Biharmonic Equation: Solutions with Bounded Dirichlet Integral

We study the unique solvability of the Dirichlet problem for the biharmonic equation in the exterior of a compact set under the assumption that a generalized solution of this problem has a bounded Dirichlet integral with weight |x|^a. Depending on the value of the parameter ^a,a we prove uniqueness theorems or present exact formulas for the dimension of the solution space of the Dirichlet problem.     

Justification of the Collocation Method for a Class of Surface Integral Equations

Mathematical Notes - The application of the collocation method to the boundary integral equation of the exterior Dirichlet boundary-value problem for the Helmholtz equation is justified. In...     

Analytic solution of an exterior Neumann problem in a non‐convex domain

A new method, based on the Kelvin transformation and the Fokas integral method, is employed for solving analytically a potential problem in a non‐convex unbounded domain of ?², assuming the Neumann boundary condition. Taking advantage of the property of the Kelvin transformation to preserve harmonicity, we apply it to the present problem. In this way, the exterior potential problem is transformed to an equivalent one in the interior domain which is the Kelvin image of the original exterior one. An integral representation of the solution of the interior problem is obtained by employing the Kelvin inversion in ?² for the Neumann data and the ‘Neumann to Dirichlet’ map for the Dirichlet data. Applying next the ‘reverse’ Kelvin transformation, we finally obtain an integral representation of the solution of the original exterior Neumann problem. Copyright © 2010 John Wiley & Sons, Ltd.     

Existence of viscosity solutions for Hessian equations in exterior domains

The Perron method is used to establish the existence of viscosity solutions to the exterior Dirichlet problems for a class of Hessian type equations with prescribed behavior at infinity.     

Dirichlet–Neumann Problem for the Biharmonic Equation in Exterior Domains

Differential Equations - We study the uniqueness and the asymptotic expansion of the solution of the mixed Dirichlet–Neumann problem for the biharmonic equation in the exterior of a compact...     

Dirichlet problem for properly elliptic equations of fourth order in the exterior of an ellipse

The present paper considers the Dirichlet problem for properly elliptic equations of fourth order in the exterior of an ellipse. No restrictions on the multiplicities of the roots of the characteristic polynomial are assumed.     

Symmetry results for overdetermined problems on convex domains via Brunn–Minkowski inequalities

We consider some well-posed Dirichlet problems for elliptic equations set on the interior or the exterior of a convex domain (examples include the torsional rigidity, the first Dirichlet eigenvalue, and the electrostatic capacity), and we add an overdetermined Neumann condition which involves the Gauss curvature of the boundary. By using concavity inequalities of Brunn–Minkowski type satisfied by the corresponding variational energies, we prove that the existence of a solution implies the symmetry of the domain. This provides some new characterizations of spheres, in models going from solid mechanics to electrostatics.     

高维Dirchlet问题数值解的概率方法

本文用概率方法求得高维Dirichlet内问题和外问题在一般区域上的数值解.高维漂移布朗族对停时具有强马氏性,它在球面上的击中时和位置分布已知,再利用Dirichlet问题解的随机表达式,我们可以获得高维Dirichlet问题的数值解.     

高维Dirichlet问题数值解的概率方法

本文用概率方法求得高维Dirichlet内问题和外问题在一般区域上的数值解\bd 高维漂移布朗族对停时具有强马氏性, 它在球面上的击中时和位置分布已知, 再利用Dirichlet问题解的随机表达式, 我们可以获得高维Dirichlet问题的数值解     

Spectral inclusion and spectral exactness for PDEs on exterior domains

Using the Dirichlet to Newmann map on the inner boundary, wegive a method to identify spurious computed eigenvalues of ellipticPDEs on exterior domains. Numerical examples illustrate itseffectiveness.